Problem 29 - Advanced Engineering Mathematics

Question

In Problems 29 and 30, evaluate the determinant of the given matrix by inspection. $$ \left(\begin{array}{rrrrrr} 4 & 0 & 0 & 0 & 0 & 0 \\ 0 & -2 & 0 & 0 & 0 & 0 \\ 0 & 0 & 3 & 0 & 0 & 0 \\ 0 & 0 & 0 & -1 & 0 & 0 \\ 0 & 0 & 0 & 0 & 2 & 0 \\ 0 & 0 & 0 & 0 & 0 & 5 \end{array}\right) $$

Step-by-Step Solution

Answer

The determinant of the matrix is 240.

Step 1: Recognize the Type of Matrix

Observe that the given matrix is a diagonal matrix. A diagonal matrix has non-zero elements only on its main diagonal, running from the top left to the bottom right, and zeros elsewhere.

Step 2: Recall the Determinant Property

For a diagonal matrix, the determinant is simply the product of the diagonal elements. This is because in a diagonal matrix, the determinant calculation simplifies as all the off-diagonal elements are zero.

Step 3: List the Diagonal Elements

Write down the diagonal elements of the matrix: \(4, -2, 3, -1, 2,\) and \(5\).

Step 4: Compute the Determinant

Multiply all the diagonal elements together to find the determinant: \[\det = 4 \times (-2) \times 3 \times (-1) \times 2 \times 5.\]Calculate this step-by-step:\[4 \times (-2) = -8, \quad -8 \times 3 = -24, \quad -24 \times (-1) = 24, \quad 24 \times 2 = 48, \quad 48 \times 5 = 240.\]

Step 5: Conclude the Calculation

The determinant of the matrix is \(240\).

Key Concepts

Matrix Properties

Matrices offer a way to represent and manipulate data in mathematical terms, and they are common in areas like physics and computer science. One interesting type of matrix is the 'diagonal matrix'. In a diagonal matrix, elements not on the main diagonal are zero. This makes computations simpler and reveals certain properties. Some essential matrix properties to remember include:

• Zero Elements: Only the diagonal can have non-zero values, meaning a diagonal matrix is always easy to spot.
• Symmetry: If a matrix equals its transpose, it is symmetric. Diagonal matrices can be symmetric if all diagonal elements are equal.
• Multiplicative Identity: When multiplying with another matrix, a diagonal matrix can act similarly to the identity matrix, preserving multiplication properties.

Diagonal matrices are simpler to use in calculations, providing a strong foundation for more complex operations in linear algebra.

Matrix Determinant

The determinant is an important scalar value in linear algebra that captures unique properties of a matrix. Though calculating the determinant for arbitrary matrices can be complex, diagonal matrices simplify this task significantly. Understanding determinant in the context of diagonal matrices involves:

• Definition: The determinant is the product of the diagonal elements in a diagonal matrix. This is because off-diagonal elements do not contribute to the determinant.
• Interpretation: The determinant can provide information about the matrix's invertibility. If the determinant is zero, the matrix is not invertible.
• Application: Determinants are used in calculating areas, volumes, and transformations in higher dimensions.

Although determinants have many uses, the simplest and most direct application is probably indicating whether a matrix can be inverted.

Linear Algebra

Linear algebra is the branch of mathematics concerning vector spaces and linear mappings. It is a backbone for many advanced mathematical concepts and practical applications, from engineering to computer graphics. Key ideas in linear algebra include:

• Vectors and Spaces: Vectors are objects that can be added together and multiplied by scalars. They exist in vector spaces, which are central to linear algebra.
• Matrix Operations: Matrices represent linear transformations. Operations like addition, multiplication, and finding determinants or inverses are vital in understanding the relationships within the space.
• Applications: Linear algebra is used in solving systems of linear equations, computer simulations, optimizations, and modeling real-world problems.

Having a solid grasp of linear algebra helps in understanding the connections between different areas of mathematics and their applications in solving practical problems.